Problem: A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate.  All of the chocolate is then evenly distributed between 27 congruent, smaller hemisphere-shaped molds.  What is the radius of each of the smaller molds, in feet?
Explanation: A sphere with radius $r$ has volume $\frac{4}{3}\pi r^3$, so a hemisphere with radius $r$ has volume $\frac{2}{3}\pi r^3$.  The large hemisphere-shaped bowl has volume $\frac{2}{3}\pi(1^3) = \frac{2}{3}\pi$ cubic feet.

Let each of the smaller hemisphere-shaped molds have radius $r$.  Their total volume, in terms of $r$, is $27\cdot\frac{2}{3}\pi r^3$ cubic feet, so we have \[27\cdot\frac{2}{3}\pi r^3 = \frac{2}{3}\pi.\]Dividing both sides by $\frac{2}{3}\pi$ yields $27r^3 =1$, so $r=\sqrt[3]{\frac{1}{27}}=\boxed{\frac{1}{3}}$ feet.